Theorem onsucuni2 3148

Description: A successor ordinal is the successor of its union.

Assertion

Ref	Expression
onsucuni2	|- ((A e. On /\ A = suc B) -> suc U.A = A)

Proof of Theorem onsucuni2

Step	Hyp	Ref	Expression
1		eleq1 1581	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
2		sucelon 3125	. . . . . 6 |- (B e. On <-> suc B e. On)
3	1, 2	syl6bbr 549	. . . . 5 |- (A = suc B -> (A e. On <-> B e. On))
4	3	biimpac 427	. . . 4 |- ((A e. On /\ A = suc B) -> B e. On)
5		eloni 3015	. . . . . . . . . . 11 |- (B e. On -> Ord B)
6		ordirr 3023	. . . . . . . . . . 11 |- (Ord B -> -. B e. B)
7	5, 6	syl 10	. . . . . . . . . 10 |- (B e. On -> -. B e. B)
8		eleq2 1582	. . . . . . . . . . 11 |- (suc B = B -> (B e. suc B <-> B e. B))
9		sucidg 3109	. . . . . . . . . . 11 |- (B e. On -> B e. suc B)
10	8, 9	syl5cbi 216	. . . . . . . . . 10 |- (B e. On -> (suc B = B -> B e. B))
11	7, 10	mtod 114	. . . . . . . . 9 |- (B e. On -> -. suc B = B)
12		ordunisuc 3146	. . . . . . . . . . 11 |- (Ord B -> U.suc B = B)
13	5, 12	syl 10	. . . . . . . . . 10 |- (B e. On -> U.suc B = B)
14	13	eqeq2d 1533	. . . . . . . . 9 |- (B e. On -> (suc B = U.suc B <-> suc B = B))
15	11, 14	mtbird 727	. . . . . . . 8 |- (B e. On -> -. suc B = U.suc B)
16	15	adantl 397	. . . . . . 7 |- ((A = suc B /\ B e. On) -> -. suc B = U.suc B)
17		id 59	. . . . . . . . 9 |- (A = suc B -> A = suc B)
18		unieq 2564	. . . . . . . . 9 |- (A = suc B -> U.A = U.suc B)
19	17, 18	eqeq12d 1536	. . . . . . . 8 |- (A = suc B -> (A = U.A <-> suc B = U.suc B))
20	19	adantr 398	. . . . . . 7 |- ((A = suc B /\ B e. On) -> (A = U.A <-> suc B = U.suc B))
21	16, 20	mtbird 727	. . . . . 6 |- ((A = suc B /\ B e. On) -> -. A = U.A)
22	21	ex 380	. . . . 5 |- (A = suc B -> (B e. On -> -. A = U.A))
23	22	adantl 397	. . . 4 |- ((A e. On /\ A = suc B) -> (B e. On -> -. A = U.A))
24	4, 23	mpd 26	. . 3 |- ((A e. On /\ A = suc B) -> -. A = U.A)
25		eloni 3015	. . . . 5 |- (A e. On -> Ord A)
26		orduniorsuc 3144	. . . . . 6 |- (Ord A -> (A = U.A \/ A = suc U.A))
27	26	ord 239	. . . . 5 |- (Ord A -> (-. A = U.A -> A = suc U.A))
28	25, 27	syl 10	. . . 4 |- (A e. On -> (-. A = U.A -> A = suc U.A))
29	28	adantr 398	. . 3 |- ((A e. On /\ A = suc B) -> (-. A = U.A -> A = suc U.A))
30	24, 29	mpd 26	. 2 |- ((A e. On /\ A = suc B) -> A = suc U.A)
31	30	eqcomd 1527	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  U.cuni 2557  Ord word 3004  Oncon0 3005  suc csuc 3007

This theorem is referenced by:  rankxplim3 4776  rankxpsuc 4777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-suc 3011

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